TRANSMISSION DYNAMICS OF STOCHASTIC SVIR INFLUENZA MODELS WITH MEDIA COVERAGE

Xinhong Zhang; Zhenfeng Shi; Hao Peng

doi:10.11948/20200444

2021 Volume 11 Issue 6

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Xinhong Zhang, Zhenfeng Shi, Hao Peng. TRANSMISSION DYNAMICS OF STOCHASTIC SVIR INFLUENZA MODELS WITH MEDIA COVERAGE[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2792-2814. doi: 10.11948/20200444

Citation:

Xinhong Zhang, Zhenfeng Shi, Hao Peng. TRANSMISSION DYNAMICS OF STOCHASTIC SVIR INFLUENZA MODELS WITH MEDIA COVERAGE[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2792-2814. doi: 10.11948/20200444

TRANSMISSION DYNAMICS OF STOCHASTIC SVIR INFLUENZA MODELS WITH MEDIA COVERAGE

1.
College of Science, China University of Petroleum, Qingdao 266580, Shandong Province, China
2.
School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast Normal University, Changchun 130024, Jilin Province, China

Corresponding author: Email: zhxinhong@163.com(X. Zhang)

Abstract

Abstract

This paper focuses on the dynamical behaviors of two stochastic SVIR models with media coverage. The first system is based on system perturbation. It is shown that the transmission dynamics can be classified by a critical value $ R_0^s $. If $ R_0^s<1 $, the disease will die out. $ R_0^s>1 $ implies that the disease will persist. Furthermore, the system has an ergodic stationary distribution if $ R_0^s>1 $. The second system is based on transmission parameter perturbation. Sufficient conditions for persistence and extinction are derived. Finally, theoretical results and numerical simulations show the effect of media coverage and environmental white noise.
- SVIR epidemic model /
- media coverage /
- stationary distribution /
- extinction and persistence
MSC: 37H05, 60H10

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References

[1]	M. E. Alexander, C. Bowman, S. M. Moghadas, R. Summers, A. B. Gumel and B. M. Sahai, A vaccination model for transmission dynamics of influenza, SIAM J. Appl. Dyn. Syst., 2004, 3(4), 503-524. doi: 10.1137/030600370 CrossRef Google Scholar
[2]	Y. Cai, Y. Kang, M. Banerjee and W. Wang, A stochastic epidemic model incorporating media coverage, Commun. Math. Sci., 2015, 14(4), 893-910. Google Scholar
[3]	Centers for Disease Control and Prevention, National Center for Health Statistics. Underlying Cause of Death 1999-2019 on CDC WONDER Online Database, released in 2020. Data are from the Multiple Cause of Death Files, 1999-2019, as compiled from data provided by the 57 vital statistics jurisdictions through the Vital Statistics Cooperative Program. Accessed at http://wonder.cdc.gov/ucd-icd10.html on Jul 14, 2021 7: 05: 18 AM. Google Scholar
[4]	K. Church and X. Liu, Analysis of a SIR model with pulse vaccination and temporary immunity: Stability, bifurcation and a cylindrical attractor, Nonlinear Anal. Real World Appl., 2019, 50, 240-266. doi: 10.1016/j.nonrwa.2019.04.015 CrossRef Google Scholar
[5]	N. H. Du and N. N. Nhu, Permanence and extinction for the stochastic SIR epidemic model, J. Differ. Equations, 2020, 269(11), 9619-9652. doi: 10.1016/j.jde.2020.06.049 CrossRef Google Scholar
[6]	S. Gao, H. Ouyang and J. Nieto, Mixed vaccination strategy in SIRS epidemic model with seasonal variability on infection, Int. J. Biomath., 2011, 4(4), 473-491. doi: 10.1142/S1793524511001337 CrossRef Google Scholar
[7]	W. Guo, Q. Zhang, X. Li and W. Wang, Dynamic behavior of a stochastic SIRS epidemic model with media coverage, Math. Meth. Appl. Sci., 2018, 41(24), 5506-5525. Google Scholar
[8]	D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 2001, 43(3), 525-546. doi: 10.1137/S0036144500378302 CrossRef Google Scholar
[9]	L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equations, 2005, 217(1), 26-53. doi: 10.1016/j.jde.2005.06.017 CrossRef Google Scholar
[10]	R. Khasminskii, Stochastic Stability of Differential equations, Sijthoff and Noordhoff press, Alphen aan den Rijn, The Netherlands, 1980. Google Scholar
[11]	A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Stat. Probab. Lett., 2013, 83(4), 960-968. doi: 10.1016/j.spl.2012.12.021 CrossRef Google Scholar
[12]	R. Lipster, A strong law of large numbers for local martingales, Stochastics, 1980, 3(1-4), 217-228. doi: 10.1080/17442508008833146 CrossRef Google Scholar
[13]	M. Liu, C. Bai and Y. Jin, Population dynamical behavior of a two-predator one-prey stochastic model with time delay, Discrete Contin. Dyn. Syst., 2017, 37(5), 2513-2538. doi: 10.3934/dcds.2017108 CrossRef Google Scholar
[14]	M. Liu and M. Fan, Permanence of stochastic Lotka-Volterra systems, J. Nonlinear Sci., 2017, 27(2), 425-452. doi: 10.1007/s00332-016-9337-2 CrossRef Google Scholar
[15]	X. Liu, Y. Takeuchi and S. Iwami, SVIR epidemic models with vaccination strategies, J. Theor. Biol., 2008, 253(1), 1-11. doi: 10.1016/j.jtbi.2007.10.014 CrossRef Google Scholar
[16]	R. Liu, J. Wu and H. Zhu, Media/psychological impact on multiple outbreaks of emerging infectious disease, Comput. Math. Meth. Med., 2007, 8(3), 153-164. doi: 10.1080/17486700701425870 CrossRef Google Scholar
[17]	D. H. Nguyen, G. Yin and C. Zhu, Long-term analysis of a stochastic SIRS model with general incidence rates, SIAM J. Appl. Math., 2020, 80(2), 814-838. doi: 10.1137/19M1246973 CrossRef Google Scholar
[18]	M. Nuño, G. Chowell and A. B Gumel, Assessing the role of basic control measures, antivirals and vaccine in curtailing pandemic influenza: Scenarios for the US, UK and the Netherlands, J. R. Soc. Interface., 2006, 4(14), 505-521. Google Scholar
[19]	S. M. Salman, Memory and media coverage effect on an HIV/AIDS epidemic model with treatment, J. Comput. Appl. Math., 2021, 385, 113203. Google Scholar
[20]	Z. Shi, X. Zhang and D. Jiang, Dynamics of an avian influenza model with half-saturated incidence, Appl. Math. Comput., 2019, 355, 399-416. Google Scholar
[21]	J. M. Tchuenche, N. Dube, C. P Bhunu, R. J. Smith and C. T. Bauch, The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health, 2011, 11(S1), S5. Google Scholar
[22]	Y. Zhao, D. Jiang and D. O'Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination, Physica A, 2013, 392(20), 4916-4927. doi: 10.1016/j.physa.2013.06.009 CrossRef Google Scholar
[23]	Y. Zhao and D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 2014, 243, 718-727. Google Scholar